Optimal. Leaf size=11 \[ \frac {\log (a \cosh (x)+b)}{a} \]
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Rubi [A] time = 0.05, antiderivative size = 11, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {3160, 2668, 31} \[ \frac {\log (a \cosh (x)+b)}{a} \]
Antiderivative was successfully verified.
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Rule 31
Rule 2668
Rule 3160
Rubi steps
\begin {align*} \int \frac {1}{a \coth (x)+b \text {csch}(x)} \, dx &=i \int \frac {\sinh (x)}{i b+i a \cosh (x)} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{i b+x} \, dx,x,i a \cosh (x)\right )}{a}\\ &=\frac {\log (b+a \cosh (x))}{a}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 11, normalized size = 1.00 \[ \frac {\log (a \cosh (x)+b)}{a} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.42, size = 27, normalized size = 2.45 \[ -\frac {x - \log \left (\frac {2 \, {\left (a \cosh \relax (x) + b\right )}}{\cosh \relax (x) - \sinh \relax (x)}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 19, normalized size = 1.73 \[ \frac {\log \left ({\left | a {\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 51, normalized size = 4.64 \[ -\frac {\ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{a}-\frac {\ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{a}+\frac {\ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b +a +b \right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.31, size = 26, normalized size = 2.36 \[ \frac {x}{a} + \frac {\log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.09, size = 23, normalized size = 2.09 \[ -\frac {x-\ln \left (a+2\,b\,{\mathrm {e}}^x+a\,{\mathrm {e}}^{2\,x}\right )}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{a \coth {\relax (x )} + b \operatorname {csch}{\relax (x )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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